On a theorem of Rickards

TL;DR

This paper reinterprets Rickards' theorem on the generating series of intersection numbers as a Kudla-Millson theta series and proves it is a diagonal restriction of a Hilbert modular form, linking it to broader modular form theories.

Contribution

It provides a new interpretation of Rickards' generating series as a Hilbert modular form, extending previous results and connecting different areas of modular form theory.

Findings

Rickards' generating series is a Kudla-Millson theta series.

The series is the diagonal restriction of a Hilbert modular form.

Links to results of Darmon-Pozzi-Vonk and Branchereau.

Abstract

A recent result of Rickards states that the generating series of intersection numbers of real quadratic geodesics on indefinite Shimura curves are elliptic modular forms. We reinterpret this as a Kudla-Millson theta series, and prove that Rickards' generating series is the diagonal restriction of a Hilbert modular form, analogous to results of Darmon-Pozzi-Vonk and Branchereau.